Problem: In a certain hyperbola, the center is at $(2,0),$ one focus is at $(2,6),$ and one vertex is at $(2,-3).$  The equation of this hyperbola can be written as
\[\frac{(y - k)^2}{a^2} - \frac{(x - h)^2}{b^2} = 1.\]Find $h + k + a + b.$
The center of the hyperbola is $(h,k) = (2,0).$  The distance between the center and one vertex is $a = 3,$ and the distance between the center and one focus is $c = 6.$  Then $b^2 = c^2 - a^2 = 6^2 - 3^2 = 27,$ so $b = 3 \sqrt{3}.$

Therefore, $h + k + a + b = 2 + 0 + 3 + 3 \sqrt{3} = \boxed{3 \sqrt{3} + 5}.$